Characteristic initial value problem for nonlinear wave equation with singular initial data

Abstract

In this paper, we study the characteristic initial value problem for a class of nonlinear wave equations with data on a conic light cone in the Minkowski space $\mathbb{R}^{1+3}$. We show the existence of local solution for a class of singular initial data in the sense that the standard energy could be infinite and the solution may blow up at the conic point. As an application, we improve our previous result on the inverse scattering problem for the Maxwell-Klein-Gordon equations with scattering data on the future null infinity.